Finding Position from Velocity or Acceleration

Exercises 41–44 give the velocity v = ds/dt and initial position of an object moving along a coordinate line. Find the object’s position at time t.

v = sin πt, s(0) = 0

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To find the position function s(t), we need to integrate the velocity function v(t) = sin(πt) with respect to time t. This is because velocity is the derivative of position, so integrating velocity will give us the position.

Set up the integral: ∫sin(πt) dt. This integral will give us the position function s(t) up to a constant of integration, which we will determine using the initial condition.

Perform the integration: The integral of sin(πt) with respect to t is -1/π * cos(πt) + C, where C is the constant of integration.

Use the initial condition s(0) = 0 to find the constant C. Substitute t = 0 into the position function: s(0) = -1/π * cos(0) + C = 0. Since cos(0) = 1, this simplifies to -1/π + C = 0, which means C = 1/π.

Substitute the constant C back into the position function to get the final expression for s(t): s(t) = -1/π * cos(πt) + 1/π.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Integration

Integration is the process of finding the antiderivative or the area under a curve. In this context, it is used to determine the position function from the velocity function. By integrating the velocity function v(t) = sin(πt), we can find the position function s(t), which describes the object's position over time.

Initial Conditions

Initial conditions are values that specify the state of a system at a particular time, often used to solve differential equations. Here, the initial condition s(0) = 0 indicates the object's position at time t = 0. This information is crucial for determining the constant of integration when finding the position function from the velocity function.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, showing that they are inverse processes. It states that if a function is continuous over an interval, the integral of its derivative over that interval equals the change in the function's values. This theorem is applied to find the position function from the velocity function by integrating and using the initial condition to solve for any constants.

Finding Position from Velocity or AccelerationExercises 41–44 give the velocity v = ds/dt and initial position of an object moving along a coordinate line. Find the object’s position at time t.v = sin πt, s(0) = 0